Throughout this paper \(\mathcal {M}\subseteq {\mathcal {B}}({\mathcal {H}})\) 
will denote a semifinite von Neumann algebra on some Hilbert space \({\mathcal {H}}\)  (here, \({\mathcal {B}}({\mathcal {H}})\)  is the algebra of all
bounded linear operators on \({\mathcal {H}}\)  equipped with the
operator norm denoted by \(\Vert \cdot \Vert _\infty \)  or
\(\Vert \cdot \Vert _{{\mathcal {B}}({\mathcal {H}})}\) ). Unless otherwise
stated, it will be assumed throughout that \(\mathcal {M}\)  is equipped with a
fixed faithful normal semifinite trace \(\tau \) . For standard facts
concerning von Neumann algebras,
we refer to [1]}, [2]}.
The identity in \(\mathcal {M}\)   is denoted
by \({\textbf {1}}\)  and we denote by \({\mathcal {P}}\left({\mathcal {M}}\right)\)  the complete lattice of all (self-adjoint) projections
in \(\mathcal {M}\) . A linear operator \(x:{\mathcal {D}}(x)\rightarrow {\mathcal {H}} \) ,
with domain \({\mathcal {D}}(x)\subseteq {\mathcal {H}}\) , is said to be
affiliated with \(\mathcal {M}\)   if \(yx\subseteq xy\)  for all \(y\)  in the commutant
\(\mathcal {M}^\prime \)  of \(\mathcal {M}\)  (equivalently, \(ux=xu\)  for all unitary \(u\)  in
\(\mathcal {M}^\prime \) ). . For any self-adjoint operator
\(x\)  on \(\mathcal {H}\) , its spectral measure is
denoted by \(e^{x}\) . A self-adjoint operator \(x\)  is affiliated with \(\mathcal {M}\)  if and only if \(e^{x}\left( B\right) \in \mathcal {P}\left( \mathcal {M}\right) \)  for any Borel set \(B\subseteq \mathbb {R}\) .
